Second derivative of parametric equation

First derivative

Given a parametric equation:

x = f(t) , y = g(t)

It is not difficult to find the first derivative by the formula:

Example 1

If x = t + cos t

y = sin t

find the first derivative.

Solution

Second derivative

However it is not true to write the formula of the second derivative as the first derivative,

that is,

Example 2

If x = t + cos t

y = sin t

find the second derivative.

Analysis

Solution

For more serious learner

The formula for the second derivative of parametric equation is given by:

Can you prove this by following the steps in Example 2?